y â k
Algebra II
Review
Functions Notation: f(x) “take whatever is in parentheses and put it in for x” Domain: input values, x-‐values Range: output values, y-‐values Vertical Line Test: x-‐values do not repeat therefore function Horizontal Line Test: determines one-‐to-‐one and whether or not the function’s inverse is a function One-‐to-‐one: x and y values do not repeat Onto: entire range is used Even: f(-‐x) = f(x), symmetry over y-‐axis Odd: f(-‐x) = -‐f(x), symmetry through the origin, looks the same upside down Inverses: Switch x and y, then solve for new y; reflection in y = x Piecewise Functions: Defined differently on specific intervals Transformations: y = a f(b(x ± c)) ± d a < 0, reflection in the x-‐axis
b < 0, reflection in the y-‐axis
c < 0, phase shift right c
|a| > 1, vertical stretch
|b| > 1, horizontal compression of 1/|b|
d > 0, up d
|a| < 1, vertical compression
d < 0, down
|b| < 1, horizontal stretch of 1/|b|
c > 0, phase shift left c
Regressions: Stat, Edit, Calc, pick regression and compare correlation coefficient, r if linear, if not r2
0 ≤ r ≤ 1 Average Rate of Change =
! ! !!(!) !!!
∆!
! !!
= slope = ∆! = !! !!! !
!
Graphs and “Shells” Constant Function
Linear Function
y = c y = mx + b or y – y1 = m(x – x1)
Absolute Value y = ___|x____|___
Quadratic {Even Polynomial}
Circle {not a function}
Cubic {Odd Polynomial}
y = ___(x___)2 + ___
(x – h)2 + (y – k)2 = r2
y = ___(x___)3 + ___
Even Radical
Odd Radical
y = ___ !"!# 𝑥___ + ___
y = ___ !"" 𝑥___ + ___
Rational Function 𝑦 = ___
! !___
+ ___
Or y = __ 𝑥___
! !"!#
+ _____ Or y = __ 𝑥___
! !""
+ _____
Exponential Function
Logarithmic Function
y = ___a__(x___) + ___
y = ___loga(x____) + ___
Cosine Function
Tangent Function
Growth Decay Sine Function
y = ___sin(___(x___)) + ___
y = ___cos(___(x___)) + ___ y = ___tan(___(x___)) + ___
Systems of Equations Linear, Linear
Linear, Quadratic, Circle
3 Equations, 3 Variables
Remember to use matrices to check your answer! [A]-‐1[B]
Quadratic Functions Standard Form:
Factored Form
f(x) = ax2 + bx + c
f(x) = a(x – r1)(x – r2)
aos: 𝑥 =
!!
!! !!!
Vertex Form
f(x) = a(x – h)2 + k
aos: 𝑥 = ℎ
y-‐int: (0, f(0))
vertex: (h, k)
aos: 𝑥 =
y-‐int: (0, c)
y-‐int: (0, f(0))
x-‐int: (r1, 0) (r2, 0)
vertex:
x-‐int: 𝑥 = vertex:
!!
!!± ! ! !!!"
!!
!!
, 𝑓 !!
!! !!
!
!! !!! !
, 𝑓
!! !!! !
Locus definition of a parabola: the set of points equidistant from a fixed point, focus, and a fixed line, directrix. p = distance between the focus and the directrix, also the distance between the focus and the vertex, 4p = focal width p > 0 p < 0
p > 0 p < 0
(x – h)2 = 4p(y – k)
vertex: (h, k)
(y – k)2 = 4p(x – h)
vertex = (h, k)
focus: (h, k + p)
directrix: y = k – p
focus: (h + p, k) directrix: x = h – p
aos: x = h aos: y = k
Solving Quadratics:
1) Set equation = 0 2) Factor, Complete the Square, or Quadratic Formula
Circle:
(x – h)2 + (y – k)2 = r2
x2 + y2 = r2
Center: (h, k)
Center: (0, 0)
Radius = r
Radius = r
Powers of i: 𝒊 = −𝟏 yi (imaginary) i0 = 1, i1 = i, i2 = -‐1, i3 = -‐I, … conjugate of a + bi = a -‐ bi
x (real)
Factoring: What do they have in common? Difference of two perfect squares, the factors are conjugate pairs: x2 – y2 = (x + y)(x – y) Sum of two perfect squares, the factors are imaginary conjugate pairs: x2 + y2 = (x + yi)(x – yi) Four Square Method Grouping: split, factor, factor, factor Chunking: u-‐substitution Sum of two perfect cubes: (x3 + y3) = (x + y)(x2 – xy + y2) Difference of two perfect cubes: (x3 -‐ y3) = (x -‐ y)(x2 + xy + y2) Perfect Trinomial Square: (x + y)2 = x2 + 2xy + y2 and (x -‐ y)2 = x2 -‐ 2xy + y2 Given Roots, find equation: x2 – (sum)x + product = 0 Complete the square: 1) To rewrite quadratic f(x) = ax2 + bx + c in vertex form 2) To solve a quadratic equation ax2 + bx + c = 0 3) To rewrite ax2 + bx + cy + d = 0 or ay2 + by + c x + d = 0 in vertex form 4) To rewrite circle equation, ax2 + ay2 + bx + cy + d = 0 in standard for
Nth Degree Polynomials Even Degree Polynomials
Odd Degree Polynomials
End Behavior: same
End Behavior: opposite
Extrema: minimum # = 1 Extrema: minimum # = 0 maximum # = n – 1 maximum # = n – 1 Global vs. Local Extrema: Global is for entire function, Local refers to all turning points
Always use the x-‐values when determining intervals where the function is increasing or decreasing.
If (x – c) is a factor of the polynomial, x = c is a root of the polynomial. Use your calculator to find the rational roots then use long division and the quadratic formula to find the others
Factor Theorem: If f(c) = 0, then (x – c) is a factor of the polynomial. Remainder Theorem: Since p(x) = (x – c)Q(x) + R, p(c) = R In other words, when a polynomial is divided by (x – c), the remainder equals p(c). Multiplicity of Roots: even when graph is tangent to the x-‐axis, odd when crossing the x-‐axis Pythagorean Triples: Start with any two positive numbers; Add their squares, Subtract their squares, Double their product Rational Functions Simplifying/Multiplying/Dividing: Factor all numerators and denominators then Bing! Remember, when you divide, first you have to flip it then multiply. Adding/Subtracting Rational Expressions: Put each “piece” over the LCD, adjust the numerators and combine like terms. “If there is no equal sign, keep it a fraction all the time.” Rational Equations: Multiply each “piece” by
!"# !
BEWARE OF EXTRANEOUS ROOTS
to clear the fractions, then solve, but…
Word Problems: Concentration
Work !
!
!
%(Amt) + %(Amt) = %(Amt) !"#$ + !"#$ = !"#$ !
!
!"!#$
Average Cost
𝐴𝑣𝑔 𝐶𝑜𝑠𝑡 =
!"#$!!"#!!"#$(!) !
d = rt
Radicals and Exponents
Solving Radical Equations:
1) Isolate the radical
2) Raise each side to the reciprocal power
3) When raising to an even root, remember the ±
4) BEWARE OF EXTRANEOUS ROOTS
Simplifying Radical Expressions:
1) Break down the number into prime factors
2) For each group of “n”, one comes out of the radical
Exponential Equations:
1) Isolate the exponential part 2) If like bases set exponents =, otherwise take ln or log of each side
Properties of Exponents: 1. Anything to the zero power = 1 2. A negative exponent means reciprocal
x0 = 1 x-‐a = 1a also, 1− a = xa
x
x
a b
3. A fractional exponent is a radical 4. When multiplying like bases, add exponents. 5. When raising an exponent to a power, multiply the exponents. xa 6. When dividing like bases, subtract exponents. = xa-‐b b x Exponential Formulas: Exponential Growth/Decay Compound Interest
x = b x a
𝑦 = 𝐴(1 ± 𝑟)!
𝑦 = 𝑃𝑒 !"
! !"
𝑦 = 𝑃 1 + !
xaxb = xa+b
(x ) = xab a b
Continuous Growth
Logarithmic Equations: 1) Use Properties of Logarithms to condense equation. 2) Exponentiate each side and solve new equation. Logarithmic Properties: 1. The log of a product is the sum of the logs. logb xy = logb x + logb y
2. The log of a quotient is the difference of the logs.
logb
logb xy = y logb x
3. The log raised to a power is the power times the log.
x = logb x -‐ logb y y
Sequences and Series
Arithmetic (common difference) Geometric (common ratio) Recursive Rule:
an = an-‐1 + d
! !! !!! !
an = (an-‐1)r
an = a1(r)n-‐1
Explicit (General) Rule: an = a1 + d(n-‐1) Sum of n terms: 𝑠! =
𝑠! =
!! !!! ! !!!
Trigonometry Function Transformations: y = A trig(B(x ± C)) ± D |A| = Amplitude |B| = Frequency C is the phase shift right or left Amplitude is the distance from a high point to the midline: 𝐴 = Frequency is the number of cycles in 2π 2π period
For sine and cosine B = for tangent =
π period
Period is the length of one cycle π 2π sine and cosine = tangent = b
b
Midline is the vertical shift up or down: 𝐷 =
!"#! ! ! !"# ! !
D is the midline
!"#! ! ! !"# ! !
More Trigonometry! SOH CAH TOA 1 degree = 60 minutes 1 minute = 60 seconds π radians = 180 degrees 180 radians → degrees multiply by π π degrees → radians multiply by 180 Special Right Triangles 30:60:90 45:45:90 𝑥: 𝑥 3: 2𝑥 𝑥: 𝑥: 𝑥 2 y y x sin θ = r cos θ = tan θ = x r Arc Length: s = r θ {θ is in radians} Unit Circle (x,y) → (cos θ ,sin θ ) S A T C
Reference Angles Quad I: Quad II: Quad III: Quad IV:
ref θ = θ ref θ = 180° -‐ θ ref θ = θ -‐ 180° ref θ = 360° -‐ θ
Probability
Venn Diagrams/Two-‐Way Tables/Hypothetical 1000 Tables Relative Frequency: Probability as a decimal Formulas: Complement {Ac, A’, 𝐴}:
Addition {OR}:
P(A U B) = P(A) + P(B) – P(A ∩ B)
P(𝐴) = 1 – P(A)
Mutually Exclusive {Disjoint}:
P(A ∩ B) = 0
Multiplication {AND}:
P(A ∩ B) = P(A) ·∙ P(B|A)
Conditional Probability:
P(A|B) =
!(! ∩ !) !(!)
Independent events: P(A ∩ B) = P(A) ·∙ P(B) Or P(A|B) = P(A)
Statistics
Sample Mean: 𝐱 Population Mean: µ Sample Standard Deviation: Sx Population Standard Deviation: σx z-‐scores: the number of standard deviations above or below the mean Find Area with Normalcdf(low, high, µ, σ) Find Scores with InvNorm(area, µ, σ) Statistical Studies: Observational (Record Observations) Survey(Questions) Experiment(Cause and Effect) Sample Proportion: 𝑝 =
# !" !"##$!!$! !
, n = sample size
Margin of Error and 95% Confidence Intervals: Population Proportion Population Mean
Statistical Significance:
1) A result outside a 95% confidence interval
2) If confidence intervals do not overlap
Review
Functions Notation: f(x) “take whatever is in parentheses and put it in for x” Domain: input values, x-‐values Range: output values, y-‐values Vertical Line Test: x-‐values do not repeat therefore function Horizontal Line Test: determines one-‐to-‐one and whether or not the function’s inverse is a function One-‐to-‐one: x and y values do not repeat Onto: entire range is used Even: f(-‐x) = f(x), symmetry over y-‐axis Odd: f(-‐x) = -‐f(x), symmetry through the origin, looks the same upside down Inverses: Switch x and y, then solve for new y; reflection in y = x Piecewise Functions: Defined differently on specific intervals Transformations: y = a f(b(x ± c)) ± d a < 0, reflection in the x-‐axis
b < 0, reflection in the y-‐axis
c < 0, phase shift right c
|a| > 1, vertical stretch
|b| > 1, horizontal compression of 1/|b|
d > 0, up d
|a| < 1, vertical compression
d < 0, down
|b| < 1, horizontal stretch of 1/|b|
c > 0, phase shift left c
Regressions: Stat, Edit, Calc, pick regression and compare correlation coefficient, r if linear, if not r2
0 ≤ r ≤ 1 Average Rate of Change =
! ! !!(!) !!!
∆!
! !!
= slope = ∆! = !! !!! !
!
Graphs and “Shells” Constant Function
Linear Function
y = c y = mx + b or y – y1 = m(x – x1)
Absolute Value y = ___|x____|___
Quadratic {Even Polynomial}
Circle {not a function}
Cubic {Odd Polynomial}
y = ___(x___)2 + ___
(x – h)2 + (y – k)2 = r2
y = ___(x___)3 + ___
Even Radical
Odd Radical
y = ___ !"!# 𝑥___ + ___
y = ___ !"" 𝑥___ + ___
Rational Function 𝑦 = ___
! !___
+ ___
Or y = __ 𝑥___
! !"!#
+ _____ Or y = __ 𝑥___
! !""
+ _____
Exponential Function
Logarithmic Function
y = ___a__(x___) + ___
y = ___loga(x____) + ___
Cosine Function
Tangent Function
Growth Decay Sine Function
y = ___sin(___(x___)) + ___
y = ___cos(___(x___)) + ___ y = ___tan(___(x___)) + ___
Systems of Equations Linear, Linear
Linear, Quadratic, Circle
3 Equations, 3 Variables
Remember to use matrices to check your answer! [A]-‐1[B]
Quadratic Functions Standard Form:
Factored Form
f(x) = ax2 + bx + c
f(x) = a(x – r1)(x – r2)
aos: 𝑥 =
!!
!! !!!
Vertex Form
f(x) = a(x – h)2 + k
aos: 𝑥 = ℎ
y-‐int: (0, f(0))
vertex: (h, k)
aos: 𝑥 =
y-‐int: (0, c)
y-‐int: (0, f(0))
x-‐int: (r1, 0) (r2, 0)
vertex:
x-‐int: 𝑥 = vertex:
!!
!!± ! ! !!!"
!!
!!
, 𝑓 !!
!! !!
!
!! !!! !
, 𝑓
!! !!! !
Locus definition of a parabola: the set of points equidistant from a fixed point, focus, and a fixed line, directrix. p = distance between the focus and the directrix, also the distance between the focus and the vertex, 4p = focal width p > 0 p < 0
p > 0 p < 0
(x – h)2 = 4p(y – k)
vertex: (h, k)
(y – k)2 = 4p(x – h)
vertex = (h, k)
focus: (h, k + p)
directrix: y = k – p
focus: (h + p, k) directrix: x = h – p
aos: x = h aos: y = k
Solving Quadratics:
1) Set equation = 0 2) Factor, Complete the Square, or Quadratic Formula
Circle:
(x – h)2 + (y – k)2 = r2
x2 + y2 = r2
Center: (h, k)
Center: (0, 0)
Radius = r
Radius = r
Powers of i: 𝒊 = −𝟏 yi (imaginary) i0 = 1, i1 = i, i2 = -‐1, i3 = -‐I, … conjugate of a + bi = a -‐ bi
x (real)
Factoring: What do they have in common? Difference of two perfect squares, the factors are conjugate pairs: x2 – y2 = (x + y)(x – y) Sum of two perfect squares, the factors are imaginary conjugate pairs: x2 + y2 = (x + yi)(x – yi) Four Square Method Grouping: split, factor, factor, factor Chunking: u-‐substitution Sum of two perfect cubes: (x3 + y3) = (x + y)(x2 – xy + y2) Difference of two perfect cubes: (x3 -‐ y3) = (x -‐ y)(x2 + xy + y2) Perfect Trinomial Square: (x + y)2 = x2 + 2xy + y2 and (x -‐ y)2 = x2 -‐ 2xy + y2 Given Roots, find equation: x2 – (sum)x + product = 0 Complete the square: 1) To rewrite quadratic f(x) = ax2 + bx + c in vertex form 2) To solve a quadratic equation ax2 + bx + c = 0 3) To rewrite ax2 + bx + cy + d = 0 or ay2 + by + c x + d = 0 in vertex form 4) To rewrite circle equation, ax2 + ay2 + bx + cy + d = 0 in standard for
Nth Degree Polynomials Even Degree Polynomials
Odd Degree Polynomials
End Behavior: same
End Behavior: opposite
Extrema: minimum # = 1 Extrema: minimum # = 0 maximum # = n – 1 maximum # = n – 1 Global vs. Local Extrema: Global is for entire function, Local refers to all turning points
Always use the x-‐values when determining intervals where the function is increasing or decreasing.
If (x – c) is a factor of the polynomial, x = c is a root of the polynomial. Use your calculator to find the rational roots then use long division and the quadratic formula to find the others
Factor Theorem: If f(c) = 0, then (x – c) is a factor of the polynomial. Remainder Theorem: Since p(x) = (x – c)Q(x) + R, p(c) = R In other words, when a polynomial is divided by (x – c), the remainder equals p(c). Multiplicity of Roots: even when graph is tangent to the x-‐axis, odd when crossing the x-‐axis Pythagorean Triples: Start with any two positive numbers; Add their squares, Subtract their squares, Double their product Rational Functions Simplifying/Multiplying/Dividing: Factor all numerators and denominators then Bing! Remember, when you divide, first you have to flip it then multiply. Adding/Subtracting Rational Expressions: Put each “piece” over the LCD, adjust the numerators and combine like terms. “If there is no equal sign, keep it a fraction all the time.” Rational Equations: Multiply each “piece” by
!"# !
BEWARE OF EXTRANEOUS ROOTS
to clear the fractions, then solve, but…
Word Problems: Concentration
Work !
!
!
%(Amt) + %(Amt) = %(Amt) !"#$ + !"#$ = !"#$ !
!
!"!#$
Average Cost
𝐴𝑣𝑔 𝐶𝑜𝑠𝑡 =
!"#$!!"#!!"#$(!) !
d = rt
Radicals and Exponents
Solving Radical Equations:
1) Isolate the radical
2) Raise each side to the reciprocal power
3) When raising to an even root, remember the ±
4) BEWARE OF EXTRANEOUS ROOTS
Simplifying Radical Expressions:
1) Break down the number into prime factors
2) For each group of “n”, one comes out of the radical
Exponential Equations:
1) Isolate the exponential part 2) If like bases set exponents =, otherwise take ln or log of each side
Properties of Exponents: 1. Anything to the zero power = 1 2. A negative exponent means reciprocal
x0 = 1 x-‐a = 1a also, 1− a = xa
x
x
a b
3. A fractional exponent is a radical 4. When multiplying like bases, add exponents. 5. When raising an exponent to a power, multiply the exponents. xa 6. When dividing like bases, subtract exponents. = xa-‐b b x Exponential Formulas: Exponential Growth/Decay Compound Interest
x = b x a
𝑦 = 𝐴(1 ± 𝑟)!
𝑦 = 𝑃𝑒 !"
! !"
𝑦 = 𝑃 1 + !
xaxb = xa+b
(x ) = xab a b
Continuous Growth
Logarithmic Equations: 1) Use Properties of Logarithms to condense equation. 2) Exponentiate each side and solve new equation. Logarithmic Properties: 1. The log of a product is the sum of the logs. logb xy = logb x + logb y
2. The log of a quotient is the difference of the logs.
logb
logb xy = y logb x
3. The log raised to a power is the power times the log.
x = logb x -‐ logb y y
Sequences and Series
Arithmetic (common difference) Geometric (common ratio) Recursive Rule:
an = an-‐1 + d
! !! !!! !
an = (an-‐1)r
an = a1(r)n-‐1
Explicit (General) Rule: an = a1 + d(n-‐1) Sum of n terms: 𝑠! =
𝑠! =
!! !!! ! !!!
Trigonometry Function Transformations: y = A trig(B(x ± C)) ± D |A| = Amplitude |B| = Frequency C is the phase shift right or left Amplitude is the distance from a high point to the midline: 𝐴 = Frequency is the number of cycles in 2π 2π period
For sine and cosine B = for tangent =
π period
Period is the length of one cycle π 2π sine and cosine = tangent = b
b
Midline is the vertical shift up or down: 𝐷 =
!"#! ! ! !"# ! !
D is the midline
!"#! ! ! !"# ! !
More Trigonometry! SOH CAH TOA 1 degree = 60 minutes 1 minute = 60 seconds π radians = 180 degrees 180 radians → degrees multiply by π π degrees → radians multiply by 180 Special Right Triangles 30:60:90 45:45:90 𝑥: 𝑥 3: 2𝑥 𝑥: 𝑥: 𝑥 2 y y x sin θ = r cos θ = tan θ = x r Arc Length: s = r θ {θ is in radians} Unit Circle (x,y) → (cos θ ,sin θ ) S A T C
Reference Angles Quad I: Quad II: Quad III: Quad IV:
ref θ = θ ref θ = 180° -‐ θ ref θ = θ -‐ 180° ref θ = 360° -‐ θ
Probability
Venn Diagrams/Two-‐Way Tables/Hypothetical 1000 Tables Relative Frequency: Probability as a decimal Formulas: Complement {Ac, A’, 𝐴}:
Addition {OR}:
P(A U B) = P(A) + P(B) – P(A ∩ B)
P(𝐴) = 1 – P(A)
Mutually Exclusive {Disjoint}:
P(A ∩ B) = 0
Multiplication {AND}:
P(A ∩ B) = P(A) ·∙ P(B|A)
Conditional Probability:
P(A|B) =
!(! ∩ !) !(!)
Independent events: P(A ∩ B) = P(A) ·∙ P(B) Or P(A|B) = P(A)
Statistics
Sample Mean: 𝐱 Population Mean: µ Sample Standard Deviation: Sx Population Standard Deviation: σx z-‐scores: the number of standard deviations above or below the mean Find Area with Normalcdf(low, high, µ, σ) Find Scores with InvNorm(area, µ, σ) Statistical Studies: Observational (Record Observations) Survey(Questions) Experiment(Cause and Effect) Sample Proportion: 𝑝 =
# !" !"##$!!$! !
, n = sample size
Margin of Error and 95% Confidence Intervals: Population Proportion Population Mean
Statistical Significance:
1) A result outside a 95% confidence interval
2) If confidence intervals do not overlap
